Analogies between entropy, volume, and cardinality inequalities for projections.

Abstract: It is well known that entropy inequalities are a quick way of obtaining volume inequalities for projections of sets in a Euclidean space (or cardinality inequalities for projections of subsets of the integer lattice) - for example, the Loomis-Whitney inequality follows easily from the classical Han’s inequality. There is also a well known connection with integral inequalities. We will review more such analogies, as well as their limitations. For example, we will observe that volume of projections to coordinate subspaces is not submodular (though entropy is), and discuss general dualities between entropy and integral inequalities. We will also discuss some particularly useful classes of Shannon-type inequalities that may be new to the AAIT audience - these also have applications to volume or cardinality. Most of this talk will be tutorial - it will be based on the work of many people in the geometry, combinatorics, probability, and information theory communities, rather than just the work of the speaker.

Computer scienceMathematics

Audience: researchers in the discipline

Seminar on Algorithmic Aspects of Information Theory

Series comments: This online seminar is a follow up of the Dagstuhl Seminar 22301, www.dagstuhl.de/en/program/calendar/semhp/?semnr=22301.